请你检查行李 is grammatically correct and I am quite sure you would be understood in a real life situation.

It is preferred to leave things implied in Chinese but only as long as there is little chance of creating confusion. When you pull away the attributive 你的 from the noun 行李, you cannot convey the specific meaning of whose luggage it is. Then you may be misunderstood as meaning luggage in general. Whether you can make this omission depends on the context.

一下 can be omitted too, but keeping it effectively makes the request softer and friendly.

I think that should be ok.

Missing the "your"?

I too noted the ambiguity. (Tons of practice trying to second-guess DL.)

Upon arrival, I would expect "examine" or "inspect."

Checked baggage is 托运行李; check-in, 值机.

You can, but sometimes it is not about the speed at all. e.g. 请您等一下can hardly be translated as Please wait quickly.
In most of the time, 一下 is referring to an action not being done thoroughly (without using full power, without paying full attention, without lasting a long time, without a large number of repetition, without doing it on a large number of objects, etc.) A (little) bit is often a good translation.